An Analytical Formulation for Optimal Dynamic Mode Decomposition

Abstract

Dynamic Mode Decomposition (DMD) is a powerful data-driven technique for model reduction that learns a low-dimensional, linear model which approximates the behavior of a high-dimensional, possibly nonlinear system. By using this low-dimensional operator, modern control theory can be applied to control the behavior of complex systems such as turbulent flows or the electrical grid. Conventional DMD finds the best linear operator for data projected onto a subspace spanned by the Proper Orthogonal Decomposition (POD) modes of the input data snapshots. These POD modes optimally capture the energy content of the data, but are not necessarily optimal for encoding system dynamics. In this talk we derive an optimal basis for the projection subspace that minimizes the error in reconstructing full state dynamics from the low-dimensional system. We show that this optimal basis constitutes an error-free projection on the span of the full state outputs. We validate the performance of the optimal projection subspace by reconstructing turbulent channel flow data and demonstrate the temporal evolution of the resulting DMD modes.

RYAN KING

National Renewable Energy Laboratory